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Inner Product Spaces
INNER PRODUCTS
Definition
An inner product on a vector space V is a function that assigns a number wv, to every pair v, w of vector
space V in such a way that the following axioms holds:
P1: wv, is a real number.
P2: vwwv,, =.
P3: uwuvuwv,,, +=+ .
P4: wvrwrv,, =.
P5: Vvvv ,0,.
Definition
A vector space V with an inner product is called an inner product space.
Note
R×VV:,.
(
)
+,,, RV is a vector space.
(
)
,,,,, +RV is an inner product space.
Examples
The following are inner product spaces.
1)
(
)
,,,,, +RR n, define YXYX=, the dot product.
2)
[
]
(
)
,,,,,, +RbaC, define
( ) ( )
=b
adxxgxfgf,.
3)
(
)
,,,,, +
×R
mn
M, define
(
)
T
ABBA tr,=.
Theorem
Let , be an inner product on a space V. Let u, v, w denote vectors in V, r a real number. Then:
1) wuvuwvu,,, +=+ .
2) wvrrwv,, =.
3) vv ,000,== .
4) 0,=vv if and only if 0
=
v.
Theorem
If A is any n×n positive definite matrix, then nTYXAYXYXR=,,, defines an inner product on Rn,
and every inner product on Rn, and every inner product on Rn arises in this way.
Proof:
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AYXYXT
=, is an inner product.
Any , on Rn can be expressed as
AY
X
T:
Let
{
}
n
EEE ,,
1= be the standard basis of Rn. Then
=
=
n
i
ii ExX
1
and
=
=
n
j
jj EyY
1
.
[ ]
AYX
y
y
EEEE
EEEE
xx
EEyxEyExYX
T
nnnn
n
n
n
ji
jiji
j
j
jj
n
i
ii
=
=
==
===
1
1
111
1
1,11
,,
,,
,,,
Moreover, T
A
A
=
.
NORMS AND DISTANCE
Definition
The norm of v in V is defined as
( )
vvvv ,norm== (length).
Definition
The distance between vectors v, w in an inner product space is
(
)
wvwv=,d.
Theorem
If 0
v is any vector in an inner product space, then v
v
v=
ˆ is the unique unit vector that is a positive
multiple of V.
Theorem: Schwarz Inequality
If v and w are two vectors in an inner product space V, then 222
,wvwv. Moreover, equality occurs if
and only if one of v or w is a scalar multiple of the other.
Proof:
Assume 0>= av and 0>= bw.
(
)
abwvwvababawbvawbvawbv==,0,2,
2, and
(
)
abwvwvababawbvawbvawbv+=++=+ ,0,2,
2.
So 22
2
2
,0,wvbawvabwvab b=.
Note: 1
,
22
2
wv
wv or 1
,
1wv
wv.
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Example
Consider the vector space
[
]
baC, of all continuous functions on
[
]
ba, . Define
( ) ( )
=b
adxxgxfgf,.
Then
( ) ( ) ( )( ) ( )( )
b
a
b
a
b
adxxgdxxfdxxgxf22
2
.
Theorem: Properties of Norms
1) 0v.
2) 0=v if an only if 0
=
v.
3) vrrv=.
4) wvwv++ (triangle inequality).
Theorem: Properties of Distance
1)
(
)
0,dwv.
2)
(
)
0,d=wv if and only if
w
v
=
.
3)
(
)
(
)
vwwv,d,d=.
4)
(
)
(
)
(
)
Vwuvwuuvwv+,,,,d,d,d
ORTHOGONAL SETS OF VECTORS
Definition
Two vectors v, w in an inner product space V are said to be orthogonal if 0,=wv.
Definition
A set of vectors
{
}
n
ee ,,
1 is called an orthogonal set if each 0
i
e and jiee ji =,0,. If, in
addition, iei=,1, then the set is called an orthonormal set.
Example
Consider
{
}
xx cos,sin in
[
]
ππ
,C with
( ) ( )
=
π
π
dxxgxfgf,. Then 0cos,sin=xx , so
{
}
xx cos,sin is an orthogonal set.
Theorem: Pythagorean Theorem
If
{
}
n
ee ,,
1 is an orthogonal set of vectors, then 22
1
2
1nn eeee ++=++ .
Theorem
Let
{
}
n
ee ,,
1 be an orthogonal set of vectors. Then:
1)
{
}
nn erer,,
11 is also orthogonal for all 0
i
r in R.
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